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New Applications of Clifford’s Geometric Algebra

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Abstract

The new applications of Clifford’s geometric algebra surveyed in this paper include kinematics and robotics, computer graphics and animation, neural networks and pattern recognition, signal and image processing, applications of versors and orthogonal transformations, spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces.

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Notes

  1. This also justifies the first title word New.

  2. We are afraid, that due to space reasons, we have not included several papers authored only by E.H.

  3. For standard references on Clifford algebra and geometric algebra, we refer to the following textbooks: [51, 84, 129]. Further in depth treatment can be found in the following textbooks: [28, 34, 80, 118, 150]. A compact definition is given in [66], see also [25].

  4. Further noteworthy references to the study of Clifford algebras are [9, 17, 38,39,40, 44, 47, 48, 110, 111, 136]. Finally, [10] provides a thorough study of the importance of Clifford algebras, on the epistemological level. We thank an anonymous reviewer for providing these important references.

  5. Even though this section appears relatively short, we remind the reader that a host of graphics relevant tools is described in other sections of this survey.

  6. For a new comprehensive textbook on quaternion and Clifford Fourier transforms, see [104]. For a framework of computer implementation of discrete quaternion and Clifford Fourier transforms, see [157].

References

  1. Abłamowicz, R., Sobczyk, G.: Appendix 7.1 of Lectures on Clifford (Geometric) Algebras and Applications. Birkhäuser, Boston, 2004

  2. Abłamowicz, R., Fauser, B.: CLIFFORD – A Maple Package for Clifford Algebra Computations with Bigebra, SchurFkt, GfG - Groebner for Grassmann, Cliplus, Define, GTP, Octonion, SP, SymGroupAlgebra, and code_support. http://www.math.tntech.edu/rafal/, December 2008

  3. Achawal, S., Lasenby, J., Hadfield, H., Lasenby, A.: Ray-Tracing Objects and Novel Surface Representations in CGA. In: Gavrilova, M., Chang, J., Magnenat-Thalmann, N., Hitzer, E., Ishikawa, H. (Eds.), Advances in Computer Graphics, 36th Computer Graphics International Conference, CGI 2019, Calgary, AB, Canada, June 17–20, 2019, Proceedings Springer International Publishing, Cham, pp. 578–584 (2019)

  4. Alho, T.: Coordinate Free Integrals in Geometric Calculus. Adv. Appl. Clifford Algebras 27, 423–437 (2017). https://doi.org/10.1007/s00006-016-0655-0

    Article  MathSciNet  MATH  Google Scholar 

  5. Alves, R., de Souza, C., Lavor, C.: Advances on the geometric algebra approach to the discretizable molecular distance geometry problem (dmdgp). In: G. Papagiannakis, D. Thalmann, P. Trahanias (Eds.), Proceedings of the 33rd Computer Graphics International, Association for Computing Machinery, New York, NY, USA, pp. 85–88 (2016)

  6. Alves, R., Lavor, C.: Geometric algebra to model uncertainties in the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebras 27(1), 439–452 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Alves, R., Hildenbrand, D., Steinmetz, C., Uftring, P.: Efficient Development of Competitive Mathematica Solutions Based on Geometric Algebra with GAALOPWeb. Adv. Appl. Clifford Algebras 30(4), 1–18 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. American Institute of Mathematics, What is \(E_8\)?, https://aimath.org/E8/e8.html, accessed 25 Feb. 2021

  9. Anglès, P.: Conformal groups in geometry and Spin structures, Birkhäuser Boston, 2008

  10. Anglès, P., Parrochia, D., Micali, A.: L’unification des Mathématiques. Hermès-Lavoisier, Cachan, France (2012)

    MATH  Google Scholar 

  11. Aouiti, C., Ben Gharbia, I.: Dynamics behavior for second-order neutral Clifford differential equations: inertial neural networks with mixed delays. Comp. Appl. Math. 39, 120 (2020). https://doi.org/10.1007/s40314-020-01148-0

    Article  MathSciNet  MATH  Google Scholar 

  12. Aristidou, A.: Hand tracking with physiological constraints. Vis Comput 34, 213–228 (2018). https://doi.org/10.1007/s00371-016-1327-8

    Article  Google Scholar 

  13. Aveneau, L., Fuchs, L., Andres, E.: Digital Geometry from a Geometric Algebra Perspective. In: Barcucci, E., Frosini, A., Rinaldi, S. (eds.) Discrete Geometry for Computer Imagery, pp. 358–369. Springer International Publishing, Cham (2014)

    MATH  Google Scholar 

  14. Belon, M.C.L., Hildenbrand, D.: Practical geometric modeling using geometric algebra motors. Adv. Appl. Clifford Algebras 27(3), 2019–2033 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Benger, W., Dobler, W.: Massive Geometric Algebra: Visions for C++ implementations of geometric algebra to scale into the big data era. Adv. Appl. Clifford Algebras 27(3), 2153–2174 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Benger, W., Hildenbrand, D., Dobler, W.: Optimizing Refined Geometric Primitive’s Leaflet Visibility for Interactive 3D Visualization via Geometric Algebra. In: N. Magnenat-Thalmann, J. Kim, H. Rushmeier, B. Levy, R. Zhang, D. Thalmann (Eds.), Proceedings of Computer Graphics International 2018. Association for Computing Machinery, New York, NY, USA, pp. 267–272 (2018)

  17. Blaine Lawson, H., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, Princeton, New Jersey (1990)

    MATH  Google Scholar 

  18. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Vol. 76 of Research Notes in Mathematics, Pitman Advanced Publishing Program, Boston, 1982

  19. Breuils, S., Nozick, V., Fuchs, L.: A geometric algebra implementation using binary tree. Adv. Appl. Clifford Algebras 27(3), 2133–2151 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Breuils, S., Nozick, V., Fuchs, L., Hildenbrand, D., Benger, W., Steinmetz, C.: A hybrid approach for computing products of high-dimensional geometric algebras. In: X. Mao, D. Thalmann, M. Gavrilova (Eds.), Proceedings of the Computer Graphics International Conference. Association for Computing Machinery, New York, NY, USA, Art. No. 43, pp. 1–6 (2017)

  21. Breuils, S., Nozick, V., Sugimoto, A., Hitzer, E.: Quadric Conformal Geometric Algebra of \({{\mathbb{R}}}^{9,6}\). Adv. Appl. Clifford Algebras 28(2), 35 (2018). https://doi.org/10.1007/s00006-018-0851-1

    Article  MATH  MathSciNet  Google Scholar 

  22. Breuils, S., Nozick, V., Fuchs, L.: Garamon: A geometric algebra library generator. Adv. Appl. Clifford Algebras 29(4), 1–41 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Breuils, S., Nozick, V., Fuchs, L., Sugimoto, A.: Transverse approach to geometric algebra models for manipulating quadratic surfaces. In: M. Gavrilova, J. Chang, N. Magnenat-Thalmann, E. Hitzer, H. Ishikawa (Eds.), Advances in Computer Graphics: 36th Computer Graphics International Conference, CGI 2019, Calgary, AB, Canada, June 17–20, 2019, Proceedings (Lecture Notes in Computer Science, 11542) Springer Nature Switzerland AG, Cham, pp. 523–534 (2019)

  24. Breuils, S., Kenmochi, Y., Sugimoto, A.: Reflexions et Rotations Digitales Bijectives avec l’Algebre Geometrique, Presentation at Journees de Geometrie Discrete et Morphologie Mathematique, Journee du GDR IGRV, 16–17 Mar. 2021 - LORIA, Villers-les-Nancy (France), https://gdmm2020.sciencesconf.org/data/GD1_BREUILS.pdf, accessed 28 Apr. 2021

  25. Breuils, S., Tachibana, K., Hitzer, E.: Introduction to Clifford’s Geometric Algebra, preprint, 10 pp., arxiv: 2108.0145, accessed 27 Aug. 2021

  26. Bujack, R., Hitzer, E., Scheuermann, G.: Demystification of the Geometric Fourier Transforms and Resulting Convolution Theorems, Math Meth Appl Sci. , Vol. 39(7), pp. 1877–1890 (2016), Article first published online: 3 Sep. 2015. https://doi.org/10.1002/mma.3607

  27. Burns, L.: Maxwell’s equations are universal for locally conserved quantities. Adv. Appl. Clifford Algebras 29, 62 (2019). https://doi.org/10.1007/s00006-019-0979-7

    Article  MathSciNet  MATH  Google Scholar 

  28. Bourbaki, N.: Elements of Mathematics, Lie Groups and Lie Algebras, Chapters 7-9, translated by Pressley, A., Springer, Berlin, 2005, Chapter 9, paragraph 9, number 1

  29. Byrtus, R., Derevianko, A., Vasik, P.: Outline of Tube Elbow Detection Based on GAC. In: N. Magnenat-Thalmann, C. Stephanidis, E. Wu, D. Thalmann, B. Sheng, J. Kim, G. Papagiannakis , M. Gavrilova (Eds.), Advances in Computer Graphics: 37th Computer Graphics International Conference, CGI 2020, Geneva, Switzerland, October 20–23, 2020, Proceedings (Image Processing, Computer Vision, Pattern Recognition, and Graphics, LNCS 12221) Springer, Cham, pp. 482–491 (2020)

  30. Chai, X., Li, Q.: Analytical mobility analysis of Bennett linkage using geometric algebra. Adv. Appl. Clifford Algebras 27(3), 2083–2095 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Cai, Z.-F., Kou, K.I.: Laplace transform: a new approach in solving linear quaternion differential equations. Math. Meth. Appl. Sci. 41, 4033–4048 (2018). https://doi.org/10.1002/mma.4415

    Article  MathSciNet  MATH  Google Scholar 

  32. Campos-Macias, L., Carbajal-Espinosa, O., Loukianov, A., Bayro-Corrochano, E.: Inverse kinematics for a 6-DOF walking humanoid robot leg. Adv. Appl. Clifford Algebras 27(1), 581–597 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Cheng, D., Kou, K.I.: Generalized sampling expansions associated with quaternion Fourier transform. Math. Meth. Appl. Sci. 41, 4021–4032 (2018). https://doi.org/10.1002/mma.4423

    Article  MathSciNet  MATH  Google Scholar 

  34. Chevalley, C.: The algebraic theory of spinors. Columbia University Press, New York (1954)

    Book  MATH  Google Scholar 

  35. Clifford, W.K.: Applications of Grassmann’s Extensive Algebra. American Journal of Mathematics 1(4), 350–358 (1878). http://www.jstor.org/stable/2369379

  36. Colapinto, P.: Spatial computing with conformal geometric algebra. Ph.D. thesis, University of California Santa Barbara (2011)

  37. Comminiello, D., Lella, M., Scardapane, S., Uncini, A.: Quaternion Convolutional Neural Networks for Detection and Localization of 3D Sound Events, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2019, pp. 8533–8537, (2019), https://doi.org/10.1109/ICASSP.2019.8682711. Preprint: arxiv: 1812.06811

  38. Crumeyrolle, A.: Algèbres de Clifford et spineurs, Université de Toulouse 3, 1974

  39. Crumeyrolle, A.: Fibrations spinorielles et twisteurs généralisés. Periodica Math. Hungarica 6–2, 143–171 (1975)

    Article  MATH  Google Scholar 

  40. Crumeyrolle, A.: Spin Fibrations over manifolds and generalized twistors. Proc. Symp. Pure Math. 27, 53–67 (1975)

    Article  MATH  Google Scholar 

  41. Da Silva, D.W., Xavier, M.A., Brown, P.N., Chow, E., de Araujo, C.P.: Homomorphic data concealment powered by Clifford geometric algebra. In: N. Magnenat-Thalmann, C. Stephanidis, E. Wu, D. Thalmann, B. Sheng, J. Kim, G. Papagiannakis , M. Gavrilova (Eds.), Advances in Computer Graphics: 37th Computer Graphics International Conference, CGI 2020, Geneva, Switzerland, October 20–23, 2020, Proceedings (Image Processing, Computer Vision, Pattern Recognition, and Graphics, LNCS 12221) Springer, Cham, pp. 513–525 (2020)

  42. Davis, A., Staples, G.S.: Zeon and Idem-Clifford Formulations of Boolean satisfiability. Adv. Appl. Clifford Algebras 29(4), 1–18 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Dechant, P.-P.: The \(E_8\) geometry from a Clifford perspective. Adv. Appl. Clifford Algebras 27, 397–421 (2017). https://doi.org/10.1007/s00006-016-0675-9

    Article  MathSciNet  MATH  Google Scholar 

  44. Deligne, P.: Notes on Spinors, in Vol. 1, Quantum Fields and Strings, A course for mathematicians, edited by P. Deligne, P. Etingof, D.S. Freed, L.C. Jeffrey, D. Kazhdan, J.W. Morgan, D.R. Morrison and E. Witten, American Mathematical Society, Providence Rhode Island, 1999

  45. De Keninck, S.: ganja.js (2020). https://doi.org/10.5281/ZENODO.3635774. URL: https://zenodo.org/record/3635774

  46. De Keninck, S., Dorst, L.: Hyperwedge. In: N. Magnenat-Thalmann, C. Stephanidis, E. Wu, D. Thalmann, B. Sheng, J. Kim, G. Papagiannakis , M. Gavrilova (Eds.), Advances in Computer Graphics: 37th Computer Graphics International Conference, CGI 2020, Geneva, Switzerland, October 20–23, 2020, Proceedings (Image Processing, Computer Vision, Pattern Recognition, and Graphics, LNCS 12221) Springer, Cham, pp. 549–554 (2020)

  47. Deheuvels, R.: Formes quadratiques et groupes classiques. P.U.F, Paris (1980)

    MATH  Google Scholar 

  48. Deheuvels, R.: Tenseurs et spineurs. P.U.F, Paris (1993)

    MATH  Google Scholar 

  49. Dong, L., Huang, L., Shao, C., et al.: Matrices of \(SL(4,{{\mathbb{R}}})\) that are the Product of Two Skew-Symmetric Matrices. Adv. Appl. Clifford Algebras 27, 475–489 (2017). https://doi.org/10.1007/s00006-016-0701-y

    Article  MathSciNet  MATH  Google Scholar 

  50. Dorst, L.: The Inner Products of Geometric Algebra. In: Dorst L., Doran C., Lasenby J. (Eds.), Applications of Geometric Algebra in Computer Science and Engineering. Birkhäuser, Boston, MA. (2002), https://doi.org/10.1007/978-1-4612-0089-5_2

  51. Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Elsevier, Burlington (2007)

    Google Scholar 

  52. Dorst, L.: Conformal Villarceau Rotors. Adv. Appl. Clifford Algebras 29(3), 1–20 (2019)

    MathSciNet  MATH  Google Scholar 

  53. Druoton, L., Fuchs, L., Garnier, L., Langevin, R.: The non-degenerate dupin cyclides in the space of spheres using geometric algebra. Adv. Appl. Clifford Algebras 24, 515–532 (2014). https://doi.org/10.1007/s00006-014-0453-5

    Article  MathSciNet  MATH  Google Scholar 

  54. Du, J., Goldman, R., Mann, S.: Modeling 3D geometry in the clifford algebra \({{\mathbb{R}}}^{4,4}\). Adv. Appl. Clifford Algebras 27(4), 3039–3062 (2017). https://doi.org/10.1007/s00006-017-0798-7

    Article  MathSciNet  MATH  Google Scholar 

  55. Dzwonkowski, M., Papaj, M., Rykaczewski, R.: A New Quaternion-Based Encryption Method for DICOM Images, in IEEE Transactions on Image Processing, vol. 24, no. 11, pp. 4614–4622, Nov. 2015 (2015), https://doi.org/10.1109/TIP.2015.2467317

  56. Easter, R.B., Hitzer, E.: Double Conformal Space-Time Algebra, S. Sivasundaram (Ed.), International Conference in Nonlinear Problems in Aviation and Aerospace ICNPAA 2016, AIP Conf. Proc., Vol. 1798, 020066 (2017); 10 pages, https://doi.org/10.1063/1.4972658, Preprint: arxiv: 1701.0651v1.pdf

  57. Easter, R.B., Hitzer, E.: Double Conformal Geometric Algebra, Adv. of App. Cliff. Algs. 27(3), pp. 2175–2199 (2017), First Online: 20th April 2017, https://doi.org/10.1007/s00006-017-0784-0, Preprint: arxiv: 1705.0019v1.pdf

  58. Easter, R.B., Hitzer, E.: Triple conformal geometric algebra for cubic plane curves. Math Meth Appl Sci. 41(11), 4088–4105 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  59. Easter, R.B., Hitzer, E.: Conic and cyclidic sections in double conformal geometric algebra \(G_{8,2}\) with computing and visualization using gaalop. Math Meth Appl Sci. 43(1), 334–357 (2019). https://doi.org/10.1002/mma.5887

    Article  MATH  MathSciNet  Google Scholar 

  60. Eelbode, D., Hitzer, E.: Operator Exponentials for the Clifford Fourier Transform on Multivector Fields in Detail, Adv. Appl. Clifford Algebras, 26(3), pp. 953-968 (2016), Online First: 22 Oct. 2015, https://doi.org/10.1007/s00006-015-0600-7, Preprint: arxiv: 1610.0244

  61. Eid, A.H.: Optimized Automatic Code Generation for Geometric Algebra Based Algorithms with Ray Tracing Application. Preprint: https://arxiv.org/abs/1607.04767v1, (2016)

  62. Eid, A.H.: An extended implementation framework for geometric algebra operations on systems of coordinate frames of arbitrary signature. Adv. Appl. Clifford Algebras 28(1), 16 (2018). https://doi.org/10.1007/s00006-018-0827-1

    Article  MathSciNet  MATH  Google Scholar 

  63. El Haoui, Y., Hitzer, E., Fahlaoui, S.: Heisenberg’s and Hardy’s uncertainty principles for special relativistic Space-Time Fourier transformation, Adv. Appl. Clifford Algebras, Sep. 2020, 30, 69, 29 pages (2020), https://doi.org/10.1007/s00006-020-01093-5

  64. El Haoui, Y., Hitzer, E.: Generalized uncertainty principles associated with the quaternionic offset linear canonical transform. Complex Variables and Elliptic Equations, published online: 28 Apr 2021, 20 pages (2021), https://doi.org/10.1080/17476933.2021.1916919

  65. Ell, T.A., Le Bihan, N., Sangwine, S.J.: Quaternion Fourier Transforms for Signal and Image Processing. Digital Signal and Image Processing, Wiley-ISTE, Hoboken (2014)

    Book  MATH  Google Scholar 

  66. Falcao, M.I., Malonek, H.R.: Generalized exponentials through Appell sets in \({{\mathbb{R}}}^{n+1}\) and Bessel functions. AIP Conf. Proc. 936, 738–741 (2007)

    Article  ADS  MATH  Google Scholar 

  67. Fontijne, D.: Gaigen 2: a geometric algebra implementation generator. GPCE ’06: Proceedings of the 5th international conference on Generative programming and component engineering, pp. 141–150 (Oct. 2006), https://doi.org/10.1145/1173706.1173728

  68. Fontijne, D.: Gaigen 2.5 User Manual (2015), https://sourceforge.net/projects/g25/, accessed: 29 Apr. 2021

  69. Franchini, S., Gentile, A., Sorbello, F., et al.: Embedded coprocessors for native execution of geometric algebra operations. Adv. Appl. Clifford Algebras 27, 559–580 (2017). https://doi.org/10.1007/s00006-016-0662-1

    Article  MathSciNet  MATH  Google Scholar 

  70. Garcia-Retuerta D., Casado-Vara R., Martin-del Rey A., De la Prieta F., Prieto J., Corchado J.M.: Quaternion Neural Networks: State-of-the-Art and Research Challenges. In: Analide C., Novais P., Camacho D., Yin H. (Eds.) Intelligent Data Engineering and Automated Learning - IDEAL 2020. Lecture Notes in Computer Science, vol 12490. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-62365-4_43

  71. Gaudet C.J., Maida A.S.: Deep Quaternion Networks, 2018 International Joint Conference on Neural Networks (IJCNN), 2018, pp. 1–8, https://doi.org/10.1109/IJCNN.2018.8489651. Preprint: arxiv: 1712.04604

  72. Goldman, R., Mann, S.: \({{\mathbb{R}}}^{4,4}\) as a computational framework for 3-dimensional computer graphics. Adv. Appl. Clifford Algebras 25(1), 113–149 (2015). https://doi.org/10.1007/s00006-014-0480-2

    Article  MathSciNet  MATH  Google Scholar 

  73. Grassmann, H.G., Kannenberg, L.C. (translator): Extension Theory (Die Ausdehnungslehre von 1862), History of Mathematics, Sources, American Mathematical Society, Rhode Island, London Mathematical Society, Volume 19 (2000)

  74. Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley and Sons, England, Chichester (1997)

    MATH  Google Scholar 

  75. Hadfield, H., Hildenbrand, D., Arsenovic, A.: Gajit: symbolic optimization and JIT compilation of geometric algebra in Python with GAALOP and Numba. In: Gavrilova, M., Chang, J., Magnenat-Thalmann, N., Hitzer, E., Ishikawa, H. (Eds.), Advances in Computer Graphics, 36th Computer Graphics International Conference, CGI 2019, Calgary, AB, Canada, June 17–20, 2019, Proceedings Springer International Publishing, Cham, pp. 499–510 (2019)

  76. Hadfield, H., Wei, L., Lasenby, J.: The Forward and Inverse Kinematics of a Delta Robot. In: N. Magnenat-Thalmann, C. Stephanidis, E. Wu, D. Thalmann, B. Sheng, J. Kim, G. Papagiannakis , M. Gavrilova (Eds.), Advances in Computer Graphics: 37th Computer Graphics International Conference, CGI 2020, Geneva, Switzerland, October 20–23, 2020, Proceedings (Image Processing, Computer Vision, Pattern Recognition, and Graphics, LNCS 12221) Springer, Cham, pp. 447–458 (2020)

  77. Hadfield, H., Achawal, S., Lasenby, J., Lasenby, A., Young, B.: Exploring novel surface representations via an experimental ray-tracer in CGA. Adv. Appl. Clifford Algebras 31(2), 1–33 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  78. Hamilton, W.R.: Elements of Quaternions, 3rd edn. Chelsea Pub Co, London (1969)

    Google Scholar 

  79. Hecht, J.P., Kamlofsky, J.A.: Hk17: Algorithm specifications and supporting documentation, Submission to NIST PQC Project (2017), Ref. [3] in [183]

  80. Helmstetter, J., Micali, A.: Quadratic Mappings and Clifford Algebras. Birkhäuser, Basel (2008)

    MATH  Google Scholar 

  81. Helmstetter, J.: Factorization of Lipschitzian Elements. Adv. Appl. Clifford Algebras 24, 675–712 (2014). https://doi.org/10.1007/s00006-014-0467-z

    Article  MathSciNet  MATH  Google Scholar 

  82. Hestenes, D.: Space Time Calculus, http://geocalc.clas.asu.edu/html/STC.html, last accessed: 17 Sep. 2020

  83. Hestenes, D.: Multivector Calculus, J. Math. Anal. and Appl., 24(2), pp. 313–325 (1968). http://geocalc.clas.asu.edu/pdf/MultCalc.pdf, last accessed: 17 Sep. 2020

  84. Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Kluwer, Dordrecht, reprinted with corrections (1992)

    MATH  Google Scholar 

  85. Hestenes, D., Li, H., Rockwood, A.: New algebraic tools for classical geometry. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebras. Springer, Berlin (2001)

    MATH  Google Scholar 

  86. Hestenes, D.: Point Groups and Space Groups in Geometric Algebra. In: Dorst, L., et al. (eds.) Applications of Geometric Algebra in Computer Science and Engineering. Birkhäuser, Basel (2002)

    Google Scholar 

  87. Hildenbrand, D., Pitt, J., Koch, A.: Gaalop—high performance parallel computing based on conformal geometric algebra. In: Bayro-Corrochano, E., Scheuermann, G. (eds.) Geometric Algebra Computing in Engineering and Computer Science, pp. 477–490. Springer, Berlin (2010)

    Chapter  MATH  Google Scholar 

  88. Hildenbrand, D., Albert, J., Charrier, P., Steinmetz, C.: Geometric algebra computing for heterogeneous systems. Adv. Appl. Clifford Algebras 27(1), 599–620 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  89. Hildenbrand, D., Franchini, S., Gentile, A., Vassallo, G., Vitabile, S.: Gappco: An easy to configure geometric algebra coprocessor based on gapp programs. Adv. Appl. Clifford Algebras 27(3), 2115–2132 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  90. Hildenbrand, D., Benger, W., Zhaoyuan, Y.: Analyzing the inner product of 2 circles with Gaalop. Math Meth Appl Sci. 41(11), 4049–4062 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  91. Hildenbrand, D., Hrdina, J., Navrat, A., Vasik, P.: Local controllability of snake robots based on CRA, theory and practice. Adv. Appl. Clifford Algebras 30(1), 1–21 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  92. Hildenbrand, D., Steinmetz, C., Alves, R., Hrdina, J., Lavor, C.: An Online Calculator for Qubits Based on Geometric Algebra. In: N. Magnenat-Thalmann, C. Stephanidis, E. Wu, D. Thalmann, B. Sheng, J. Kim, G. Papagiannakis , M. Gavrilova (Eds.), Advances in Computer Graphics: 37th Computer Graphics International Conference, CGI 2020, Geneva, Switzerland, October 20–23, 2020, Proceedings (Image Processing, Computer Vision, Pattern Recognition, and Graphics, LNCS 12221) Springer, Cham, pp. 526–537 (2020)

  93. Hildenbrand, D., Steinmetz, C., Tichy, R.: GAALOPWeb for MATLAB: An easy to handle solution for industrial geometric algebra implementations. Adv. Appl. Clifford Algebras 30(4), 1–18 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  94. Hitzer, E., Tachibana, K., Buchholz, S., Yu, I.: Carrier Method for the General Evaluation and Control of Pose, Molecular Conformation, Tracking, and the Like, Adv. in App. Cliff. Alg., 19(2), (2009) pp. 339–364. https://doi.org/10.1007/s00006-009-0160-9. Preprint: https://www.researchgate.net/publication/226288320_Carrier_Method_for_the_General_Evaluation_and_Control_of_Pose_Molecular_Conformation_Tracking_and_the_Like

  95. Hitzer, E., Perwass, C.: Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups, Adv. Appl. Clifford Alg., Vol. 20(3-4), pp. 631-658, (2010), https://doi.org/10.1007/s00006-010-0214-z, Preprint: arxiv: 1306.0158

  96. Hitzer, E., Nitta, T., Kuroe, Y.: Applications of Clifford’s geometric algebra. Adv. Appl. Clifford Algebras 23, 377–404 (2013). https://doi.org/10.1007/s00006-013-0378-4

    Article  MathSciNet  MATH  Google Scholar 

  97. Hitzer, E., Sangwine, S.J.: The orthogonal 2D planes split of quaternions and steerable quaternion Fourier transformations, in: E. Hitzer and S.J. Sangwine (Eds.), Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics (TIM) 27, Birkhäuser, 2013, pp. 15–40. https://doi.org/10.1007/978-3-0348-0603-9_2 , preprint: arxiv: 1306.2157

  98. Hitzer, E., Sangwine, S.J.: Multivector and multivector matrix inverses in real Clifford algebras, Appl. Math. and Comp., Vol. 311, Iss. C, Oct. 2017, pp. 375–389 (2017), https://doi.org/10.1016/j.amc.2017.05.027. Preprint: Technical Report CES-534, ISSN: 1744-8050, http://repository.essex.ac.uk/17282

  99. Hitzer, E., Hildenbrand, D.: Cubic curves and cubic surfaces from contact points in conformal geometric algebra. In: Gavrilova, M., Chang, J., Magnenat-Thalmann, N., Hitzer, E., Ishikawa, H. (Eds.), Advances in Computer Graphics, 36th Computer Graphics International Conference, CGI 2019, Calgary, AB, Canada, June 17–20, 2019, Proceedings Springer International Publishing, Cham, pp. 535–545 (2019)

  100. Hitzer, E., Sangwine, S.J.: Foundations of conic conformal geometric algebra and compact versors for rotation, translation and scaling. Adv. Appl. Clifford Algebras 29, 96 (2019). https://doi.org/10.1007/s00006-019-1016-6

    Article  MathSciNet  MATH  Google Scholar 

  101. Hitzer, E. and Sangwine, S.J.: Construction of multivector inverse for Clifford algebras over \(2m+1\)-dimensional vector spaces from multivector inverse for Clifford algebras over \(2m\)-dimensional vector spaces, Adv. of App. Cliff. Algs., (2019) 29(2):29, pp. 1–22, https://doi.org/10.1007/s00006-019-0942-7, Preprint: http://vixra.org/pdf/1901.0246v1.pdf

  102. Hitzer, E., Benger, W., Niederwieser, M., Baran, R., Steinbacher, F.: Strip Adjustment of Airborne Laserscanning Data with Conformal Geometric Algebra, 2020 Joint 11th International Conference on Soft Computing and Intelligent Systems and 21st International Symposium on Advanced Intelligent Systems (SCIS-ISIS), Publisher: IEEE, 6 pages (2020). https://doi.org/10.1109/SCISISIS50064.2020.9322694

  103. Hitzer, E., Perwass, C., Proserpio, D.M. (foreword): Space Group Visualizer, Independently published – KDP, Seattle (US), 2021, 162 pages, ISBN: 979-8719838618, URL: https://www.amazon.com/dp/B08YRYMPR5, 13 Mar. 2021

  104. Hitzer, E.: Quaternion and Clifford Fourier Transforms. Chapman and Hall/CRC, London (2021)

    Book  MATH  Google Scholar 

  105. Hrdina, J., Navrat, A., Vasik, P., Matousek, R.: CGA-based robotic snake control. Adv. Appl. Clifford Algebras 27(1), 621–632 (2017). https://doi.org/10.1007/s00006-016-0695-5

    Article  MathSciNet  MATH  Google Scholar 

  106. Hrdina, J., Matousek, R., Navrat, A., Vasik, P.: Fisheye correction by CGA non-linear transformation. Math Meth Appl Sci. 41, 4106–4116 (2018). https://doi.org/10.1002/mma.4455

    Article  MathSciNet  MATH  Google Scholar 

  107. Hrdina, J., Vasik, P., Matousek, R., Navrat, A.: Geometric algebras for uniform color spaces. Math Meth Appl Sci. 41, 4117–4130 (2018). https://doi.org/10.1002/mma.4489

    Article  MATH  Google Scholar 

  108. Hrdina, J., Navrat, A., Vasik, P.: Geometric algebra for conics. Adv. Appl. Clifford Algebras 28(66), 1–21 (2018). https://doi.org/10.1007/s00006-018-0879-2

    Article  MathSciNet  MATH  Google Scholar 

  109. Hrdina, J., Navrat, A., Vasik, P.: Conic fitting in geometric algebra setting. Adv. Appl. Clifford Algebras 29(4), 1–13 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  110. Karoubi M.: Algèbres de Clifford et K-théorie, Annales scientifiques de l’E.N.S., quatrième série, tome 1, pp. 14–270 (1964)

  111. Karoubi, M.: K-theory, an introduction. Springer-Verlag, Berlin (1978)

    Book  MATH  Google Scholar 

  112. Kobayashi, M.: Quaternion projection rule for rotor Hopfield neural networks. IEEE Trans. Neural Netw. Learn. Syst. 32(2), 900–908 (2021). https://doi.org/10.1109/TNNLS.2020.2979920

    Article  MathSciNet  Google Scholar 

  113. Kobayashi, M.: Quaternion-valued twin-multistate Hopfield neural networks with dual connections. IEEE Trans. Neural Netw. Learn. Syst. 32(2), 892–899 (2021). https://doi.org/10.1109/TNNLS.2020.2979904

    Article  MathSciNet  Google Scholar 

  114. Kobayashi, M.: Hybrid quaternionic Hopfield neural network, The Institute of Electronics, Information and Communication Engineers Trans. Fundamentals E98.A(7), pp. 1512–1518, (2015). URL: http://ci.nii.ac.jp/naid/130005085803

  115. Kobayashi, M.: Synthesis of complex- and hyperbolic-valued Hopfield neural networks, Neurocomputing, Volume 423, pp. 80–88, 29 January 2021, https://doi.org/10.1016/j.neucom.2020.10.002

  116. Kobayashi, M., et al.: Hyperbolic-Valued Hopfield Neural Networks in Hybrid Mode, Neurocomputing Volume 440, pp. 275–278 (2021). https://doi.org/10.1016/j.neucom.2021.01.121

  117. Krasauskas, R.: Unifying theory of pythagorean-normal surfaces based on geometric algebra. Adv. Appl. Clifford Algebras 27(1), 491–502 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  118. Lam, T.Y.: The algebraic theory of quadratic forms. W.A. Benjamin Inc, New York (1973). (Chapter 5)

  119. Lasenby, A.N.: Geometric algebra as a unifying language for physics and engineering and its use in the study of gravity. Adv. Appl. Clifford Algebras 27, 733–759 (2017). https://doi.org/10.1007/s00006-016-0700-z

    Article  MathSciNet  MATH  Google Scholar 

  120. Lasenby, A.N.: Geometric algebra, gravity and gravitational waves. Adv. Appl. Clifford Algebras 29, 79 (2019). https://doi.org/10.1007/s00006-019-0991-y

    Article  MathSciNet  MATH  Google Scholar 

  121. Leao, R.d.F., Wainer, S.A.: Immersion in \({{\mathbb{S}}}^n\) by complex spinors. Adv. Appl. Clifford Algebras 29, 65 (2019). https://doi.org/10.1007/s00006-019-0986-8

  122. Leopardi, P.C.: Gastineau-Hills’ quasi-Clifford algebras and plug-in constructions for Hadamard matrices. Adv. Appl. Clifford Algebras 29, 48 (2019). https://doi.org/10.1007/s00006-019-0963-2

    Article  MathSciNet  MATH  Google Scholar 

  123. Lewintan, P.: Geometric Calculus of the Gauss Map. Adv. Appl. Clifford Algebras 27, 503–521 (2017). https://doi.org/10.1007/s00006-016-0727-1

    Article  MathSciNet  MATH  Google Scholar 

  124. Li, B., Li, Y.: Existence and global exponential stability of pseudo almost periodic solution for Clifford-Valued neutral high-order Hopfield neural networks with leakage delays, in IEEE Access, vol. 7, pp. 150213–150225, 2019, https://doi.org/10.1109/ACCESS.2019.2947647, Open Access

  125. Li, Y., Xiang, J.: Existence and global exponential stability of anti-periodic solution for Clifford-valued inertial Cohen-Grossberg neural networks with delays, Neurocomputing, Volume 332, pp. 259–269, 7 March 2019, URL: https://www.sciencedirect.com/science/article/pii/S0925231218315261, https://doi.org/10.1016/j.neucom.2018.12.064

  126. Li, Y., Huo, N., Li, B.: On \(\mu \)-Pseudo Almost Periodic Solutions for Clifford-Valued Neutral Type Neural Networks With Delays in the Leakage Term, in IEEE Transactions on Neural Networks and Learning Systems, vol. 32, no. 3, pp. 1365–1374, March 2021, https://doi.org/10.1109/TNNLS.2020.2984655. URL: https://ieeexplore.ieee.org/document/9067056

  127. Liu, Y., Zheng, Y., Lu, J., Cao, J., Rutkowski, L.: Constrained quaternion-variable convex optimization: A quaternion-valued recurrent neural network approach. IEEE Trans. Neural Netw. Learn. Syst. 31(3), 1022–1035 (2020). https://doi.org/10.1109/TNNLS.2019.2916597

    Article  MathSciNet  Google Scholar 

  128. Liu, Y., Zhang, D., Lou, J., Lu, J., Cao, J.: Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches, Published in: IEEE Transactions on Neural Networks and Learning Systems (Volume: 29, Issue: 9 , Sept. 2018) URL: https://ieeexplore.ieee.org/document/8088357

  129. Lounesto, P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (UK) (2001)

    Book  MATH  Google Scholar 

  130. Lu, J., Hu, D., Wang, T., Zhang, Z.: GASDL: Geometric algebra-based spatial data description Language. In: N. Magnenat-Thalmann, J. Kim, H. Rushmeier, B. Levy, R. Zhang, D. Thalmann (Eds.), Proceedings of Computer Graphics International 2018. Association for Computing Machinery, New York, NY, USA, pp. 261–265 (2018)

  131. Luo, W., Hu, Y., Yu, Z., Yuan, L., Lu, G.: A hierarchical representation and computation scheme of arbitrary-dimensional geometrical primitives based on CGA. Adv. Appl. Clifford Algebras 27(3), 1977–1995 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  132. Luo, W., Li, D., Yu, Z., Wang, Y., Yan, Z., Yuan, L.: Geometric Algebra-Based Multilevel Declassification Method for Geographical Field Data. In: N. Magnenat-Thalmann, C. Stephanidis, E. Wu, D. Thalmann, B. Sheng, J. Kim, G. Papagiannakis , M. Gavrilova (Eds.), Advances in Computer Graphics: 37th Computer Graphics International Conference, CGI 2020, Geneva, Switzerland, October 20–23, 2020, Proceedings (Image Processing, Computer Vision, Pattern Recognition, and Graphics, LNCS 12221) Springer, Cham, pp. 501–512 (2020)

  133. McClellan, G.E.: Application of geometric algebra to the electroweak sector of the standard model of particle physics. Adv. Appl. Clifford Algebras 27, 761–786 (2017). https://doi.org/10.1007/s00006-016-0685-7

    Article  MathSciNet  MATH  Google Scholar 

  134. McClellan, G.E.: Using raising and lowering operators from geometric algebra for electroweak theory in particle physics. Adv. Appl. Clifford Algebras 29, 90 (2019). https://doi.org/10.1007/s00006-019-1002-z

    Article  MathSciNet  MATH  Google Scholar 

  135. Meister, L., Schaeben, H.: A concise quaternion geometry of rotations. MMAS 28, 101–126 (2005). https://doi.org/10.1002/mma.560

    Article  ADS  MathSciNet  MATH  Google Scholar 

  136. Michelson, M.-L.: Clifford and spinor cohomology. Am. J. Math. 106(6), 1083–1146 (1980)

    Article  MathSciNet  Google Scholar 

  137. Mustard, D.: Fractional convolution. J. Aust. Math. Soc. Ser. B 40, 257–265 (1998). https://doi.org/10.1017/S0334270000012509

    Article  MathSciNet  MATH  Google Scholar 

  138. Ong, J.: Klein—Geometric Algebra Library for C++, http://www.jeremyong.com/klein/, accessed 28 Apr. 2021

  139. Orouji, N., Sadr, A.: A Hardware Implementation for Color Edge Detection Using Prewitt-Inspired Filters Based on Geometric Algebra. Adv. Appl. Clifford Algebras 29, 23 (2019). https://doi.org/10.1007/s00006-019-0941-8

    Article  MATH  Google Scholar 

  140. Papaefthymiou, M., Hildenbrand, D., Papagiannakis, G.: An inclusive conformal geometric algebra GPU animation interpolation and deformation algorithm. Vis. Comput. 32(6), 751–759 (2016)

    Article  Google Scholar 

  141. Papaefthymiou, M., Hildenbrand, D., Papagiannakis, G.: A conformal geometric algebra code generator comparison for virtual character simulation in mixed reality. Adv. Appl. Clifford Algebras 27(3), 2051–2066 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  142. Papagiannakis, G.: Geometric algebra rotors for skinned character animation blending. In SIGGRAPH Asia 2013 Technical Briefs (SA ’13). Association for Computing Machinery, New York, NY, USA, Article 11, 1-6 (2013). https://doi.org/10.1145/2542355.2542369

  143. Papagiannakis, G., Papanikolaou, P., Greassidou, E., Trahanias, P.: glGA: an OpenGL Geometric Application Framework for a Modern, Shader-based Computer Graphics Curriculum. in Bourdin, J.-J., Jorge, J., Anderson, E. (eds.), Education Papers, The Eurographics Association, Strasbourg, 8 pages (2014). Preprint: http://george.papagiannakis.org/wp-content/uploads/2014/02/glGAFrameworkDescriptionFinal.pdf

  144. Parcollet, T., Morchid, M., Linares, G.: A survey of quaternion neural networks. Artif. Intell. Rev. 53, 2957–2982 (2020). https://doi.org/10.1007/s10462-019-09752-1

    Article  Google Scholar 

  145. Parkin, S.T.: A model for quadric surfaces using geometric algebra. Unpublished, October (2012)

  146. Pavllo, D., Grangier, D. and Auli, M.: QuaterNet: A Quaternion-based Recurrent Model for Human Motion. British Machine Vision Conference, 2018, 14 pages, URL: http://www.bmva.org/bmvc/2018/contents/papers/0675.pdf, Open Access

  147. Perez-Gracia, A., Thomas, F.: On Cayley’s factorization of 4d rotations and applications. Adv. Appl. Clifford Algebras 27, 523–538 (2017). https://doi.org/10.1007/s00006-016-0683-9

    Article  MathSciNet  MATH  Google Scholar 

  148. Perwass, C.: Geometric algebra with applications in engineering, Geometry and Computing, vol. 4. Springer, Berlin (2009)

    MATH  Google Scholar 

  149. Pluta, K., Romon, P., Kenmochi, Y., Passat, N.: Bijectivity Certification of 3D Digitized Rotations. In: Computational Topology in Image Context, pp. 30–41. Springer International Publishing, Cham (2016)

  150. Porteous, I.R.: Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge (UK) (1995)

    Book  MATH  Google Scholar 

  151. Prodanov, D., Toth, V.T.: Sparse representations of Clifford and tensor algebras in maxima. Adv. Appl. Clifford Algebras 27(1), 661–683 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  152. Rajchakit, G., Sriraman, R., Lim, C.P., Unyong, B.: Existence, uniqueness and global stability of Clifford-valued neutral-type neural networks with time delays, Mathematics and Computers in Simulation, Available online 5 March 2021, https://doi.org/10.1016/j.matcom.2021.02.023

  153. Rodrigues, W.A., Wainer, S.A.: Equations of motion and energy-momentum 1-forms for the coupled gravitational, Maxwell and Dirac fields. Adv. Appl. Clifford Algebras 27, 787–803 (2017). https://doi.org/10.1007/s00006-016-0679-5

    Article  MathSciNet  MATH  Google Scholar 

  154. Roelfs, M, De Keninck, S.: Graded Symmetry Groups: Plane and Simple. 17 pp., preprint: arxiv: 2107.03771.pdf, accessed 31 Aug. 2021

  155. Romero, N., Barron-Fernandez, R., Godoy-Calderon, S.: Constructing voronoi diagrams from hollow spheres using conformal geometric algebra. Adv. Appl. Clifford Algebras 27(3), 1997–2017 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  156. Roussillon, T., Coeurjolly, D.: Characterization of bijective discretized rotations by Gaussian integers. Research report, LIRIS UMR CNRS 5205 (2016)

  157. Sangwine S.J., Todd A. Ell, T.A.: Complex and hypercomplex discrete Fourier transforms based on matrix exponential form of Euler’s formula, Applied Mathematics and Computation, Vol. 219, Iss. 2, 644–655 (2012). https://doi.org/10.1016/j.amc.2012.06.055

  158. Sangwine, S.J., Le Bihan, N.: Quaternion and octonion toolbox for Matlab, http://qtfm.sourceforge.net/, last accessed 29 Mar. 2016

  159. Sangwine, S.J., Hitzer, E.: Clifford Multivector Toolbox (for MATLAB) 2015–2016, Software library available at: http://clifford-multivector-toolbox.sourceforge.net/, last accessed 18 Sep. 2020

  160. Sangwine, S.J., Hitzer, E.: Clifford Multivector Toolbox (for MATLAB). Adv. Appl. Clifford Algebras 27, pp. 539–558 (2017). Online First 19 April 2016, https://doi.org/10.1007/s00006-016-0666-x, Preprint: http://repository.essex.ac.uk/16434/1/author_final.pdf

  161. Sangwine, S.J., Hitzer, E.: Polar decomposition of complexified quaternions and octonions. Adv. Appl. Clifford Algebras 30, 23 (2020). https://doi.org/10.1007/s00006-020-1048-y

    Article  MathSciNet  MATH  Google Scholar 

  162. Sarabandi, S., Perez-Gracia, A., Thomas, F.: On Cayley’s Factorization with an Application to the Orthonormalization of Noisy Rotation Matrices. Adv. Appl. Clifford Algebras 29, 49 (2019), https://doi.org/10.1007/s00006-019-0965-0

  163. Shao, Z., Shu, H., Wu, J., Dong, Z., Coatrieux, G., Coatrieux, J.L.: Double color image encryption using iterative phase retrieval algorithm in quaternion gyrator domain. Opt. Express 22(5), 4932–4943 (2014). https://doi.org/10.1364/OE.22.004932

    Article  ADS  Google Scholar 

  164. Shao, Z., Shang, Y., Zeng, R., Shu, H., Coatrieux, G., Wu, J.: Robust watermarking scheme for color image based on quaternion-type moment invariants and visual cryptography, Signal Processing: Image Communication, Volume 48, , Pages 12–21 (2016), ISSN 0923-5965, https://doi.org/10.1016/j.image.2016.09.001, URL: https://www.sciencedirect.com/science/article/pii/S0923596516301175

  165. Shen, S., Li, Y.: Weighted pseudo almost periodic solutions for Clifford-valued neutral-type neural networks with leakage delays on time scales. Adv Differ Equ 2020, 286 (2020). https://doi.org/10.1186/s13662-020-02754-2, Open Access

  166. Shirokov, D.S.: Concepts of trace, determinant and inverse of Clifford algebra elements, In: Proc. 8th Congress of ISAAC, edited by V. I. Burenkov et al, Vol. 1, pp. 187–194, Friendship Univ. of Russia (2012), Preprint: arXiv:1108.5447

  167. Shirokov, D.S.: Calculation of elements of spin groups using method of averaging in Clifford’s geometric algebra. Adv. Appl. Clifford Algebras 29, 50 (2019). https://doi.org/10.1007/s00006-019-0967-y

    Article  MathSciNet  MATH  Google Scholar 

  168. Shirokov, D.S.: On basis-free solution to sylvester equation in geometric algebra. In: Magnenat-Thalmann N. et al. (Eds.), Advances in computer graphics. CGI 2020. Lecture Notes in Computer Science, vol 12221. Springer, Cham. pp 541–548 (2020), https://doi.org/10.1007/978-3-030-61864-3_46

  169. Simsek, S., Sarduvan, M., Oezdemir, H.: Centrohermitian and skew-centrohermitian solutions to the minimum residual and matrix nearness problems of the quaternion matrix equation \((AXB, DXE) = (C, F)\). Adv. Appl. Clifford Algebras 27, 2201–2214 (2017). https://doi.org/10.1007/s00006-016-0688-4

    Article  MathSciNet  MATH  Google Scholar 

  170. G. Soria-Garcia, G., Altamirano-Gomez, G., Ortega-Cisneros, S., Bayro-Corrochano, E.: FPGA implementation of a geometric voting scheme for the extraction of geometric entities from images. Adv. Appl. Clifford Algebras 27(1), 685–705 (2017)

  171. Sousa, E.V., Fernandes, L.A.F.: TbGAL: a tensor-based library for geometric algebra. Adv. Appl. Clifford Algebras 30, 27 (2020). https://doi.org/10.1007/s00006-020-1053-1

    Article  MathSciNet  MATH  Google Scholar 

  172. Staples, G.S.: Zeons, orthozeons, and processes on colored graphs. In: X. Mao, D. Thalmann, M. Gavrilova (Eds.), Proceedings of the Computer Graphics International Conference. Association for Computing Machinery, New York, NY, USA, Art. No. 42, pp. 1–6 (2017)

  173. Tay Y. et. al.: Lightweight and efficient neural natural language processing with quaternion networks, in Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, Florence, Italy, July 28 – August 2, 2019, pp. 1494–1503, 2019. URL: https://www.aclweb.org/anthology/P19-1145.pdf, Open Access

  174. Tichy, R.: Application of 2D PGA as an Subalgebra of CRA in Robotics. In: N. Magnenat-Thalmann, C. Stephanidis, E. Wu, D. Thalmann, B. Sheng, J. Kim, G. Papagiannakis , M. Gavrilova (Eds.), Advances in Computer Graphics: 37th Computer Graphics International Conference, CGI 2020, Geneva, Switzerland, October 20–23, 2020, Proceedings (Image Processing, Computer Vision, Pattern Recognition, and Graphics, LNCS 12221) Springer, Cham, pp. 472–481 (2020)

  175. Tordal, S.S., Hovland, G., Tyapin, I.: Efficient implementation of inverse kinematics on a 6-dof industrial robot using conformal geometric algebra. Adv. Appl. Clifford Algebras 27(3), 2067–2082 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  176. Valle, M.E., Lobo, R.A.: Hypercomplex-valued recurrent correlation neural networks. Neurocomputing 432(7), 111–123 (2021). https://doi.org/10.1016/j.neucom.2020.12.034

    Article  Google Scholar 

  177. Vaz, J., da Rocha, R.: An Introduction to Clifford Algebras and Spinors. Oxford University Press, Oxford (UK) (2016)

    Book  MATH  Google Scholar 

  178. Vold, T.G.: Computational electromagnetism by the method of least action. Adv. Appl. Clifford Algebras 27, 805–828 (2017). https://doi.org/10.1007/s00006-016-0681-y

    Article  MathSciNet  MATH  Google Scholar 

  179. Wang, Y.: Octonion algebra and noise-free fully homomorphic encryption (FHE) scheme, Eprint URL: arXiv:1601.06744, 39 pages, Jan. 2016

  180. Wang, Y.: Notes on Two fully homomorphic encryption schemes without bootstrapping, Cryptology ePrint Archive: Report 2015/519, 6 pages (2015), URL: https://eprint.iacr.org/2015/519.pdf

  181. Wang, Y., Malluhi, Q.M.: Privacy Preserving Computation in cloud using noise-free fully homomorphic encryption (FHE) schemes, In: Askoxylakis I., Ioannidis S., Katsikas S., Meadows C. (Eds.) Computer Security – ESORICS 2016. ESORICS 2016. Lecture Notes in Computer Science, vol 9878. Springer, Cham (2016), https://doi.org/10.1007/978-3-319-45744-4_15

  182. Wang, Y., Malluhi, Q.M.: Remarks on quaternions/octonion based diffie-hellman key exchange protocol submitted to NIST PQC project, IACR Cryptol. ePrint Arch. 2017/1258, 8 pages (2017), URL: https://eprint.iacr.org/2017/1258.pdf

  183. Wang, Y., Zhang, F.: A unified CGA-based formal expression of spatio-temporal topological relations for computation and analysis of geographic objects. Adv. Appl. Clifford Algebras 29(4), 1–17 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  184. Yagisawa, M.: Fully homomorphic encryption without bootstrapping, Technical report, Cryptology ePrint Archive, Report 2015/474, (2015), URL: http://eprint.iacr.org/2015/474

  185. Yagisawa, M.: Fully homomorphic encryption on octonion ring, Cryptology ePrint Archive: Report 2015/733, 42 pages (2015), URL: https://eprint.iacr.org/2015/733.pdf

  186. Yagisawa, M.: Fully homomorphic encryption with isotropic elements, Cryptology ePrint Archive: Report 2016/462, 30 pages (2016), URL: https://eprint.iacr.org/2016/462.pdf

  187. Yao, H., Chen, Q., Chai, X., Li, Q.: Singularity analysis of 3-RPR parallel manipulators using geometric algebra. Adv. Appl. Clifford Algebras 27(3), 2097–2113 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  188. Yu, Z., Li, D., Zhu, S., Luo, W., Hu, Y., Yuan, L.: Multisource multisink optimal evacuation routing with dynamic network changes: A geometric algebra approach. Math. Meth. Appl. Sci. 41(11), 4179–4194 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  189. Yuan, S., Zhu, S., Li, D.S., Luo, W., Yu, Z.Y., Yuan, L.W.: Feature preserving multiresolution subdivision and simplification of point clouds: a conformal geometric algebra approach. Math. Meth. Appl. Sci. 41(11), 4074–4087 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  190. Zatloukal, V.: Hamiltonian constraint formulation of classical field theories. Adv. Appl. Clifford Algebras 27, 829–851 (2017). https://doi.org/10.1007/s00006-016-0663-0

    Article  MathSciNet  MATH  Google Scholar 

  191. Zhang, J., Yin, P., Wang, C., Chen, T., Shi, Z.: 3D topological error detection for cadastral parcels based on conformal geometric algebra. Adv. Appl. Clifford Algebras 29(4), 1–17 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  192. Zhang, A., Tay, Y., Zhang, S., Chan, A.T., Luu, A., Hui, S.C., Fu, J.: Beyond fully-connected layers with quaternions: parametrization of hypercomplex multiplications with \(1/n\) parameters. Published as a conference paper at the 9th International Conference on Learning Representations (ICLR 2021), 2021. Preprint: arXiv:2102.08597, accessed: 27 Apr. 2021

  193. Zou, C., Kou, K.I., Morais, J.: Prolate spheroidal wave functions associated with the quaternionic Fourier transform Math. Meth. Appl. Sci. 41, 4003–4020 (2018). https://doi.org/10.1002/mma.4439

    Article  MATH  Google Scholar 

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Acknowledgements

We thank David Da Silva for references on applications of quaternions, octonions and geometric algebra to cryptography. E.H. thanks God: Soli Deo Gloria.

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    Stephane Breuils

  2. Kogakuin University, Tokyo, Japan

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  3. International Christian University, Tokyo, Japan

    Eckhard Hitzer

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Breuils, S., Tachibana, K. & Hitzer, E. New Applications of Clifford’s Geometric Algebra. Adv. Appl. Clifford Algebras 32, 17 (2022). https://doi.org/10.1007/s00006-021-01196-7

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  • DOI: https://doi.org/10.1007/s00006-021-01196-7

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